This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A342980 #24 Jan 09 2025 15:43:42
%S A342980 1,0,0,0,1,0,0,1,1,0,0,1,8,1,0,0,1,20,20,1,0,0,1,38,131,38,1,0,0,1,63,
%T A342980 469,469,63,1,0,0,1,96,1262,3008,1262,96,1,0,0,1,138,2862,12843,12843,
%U A342980 2862,138,1,0,0,1,190,5780,42602,83088,42602,5780,190,1,0
%N A342980 Triangle read by rows: T(n,k) is the number of rooted loopless planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.
%C A342980 The number of vertices is n + 2 - k.
%C A342980 For k >= 2, columns k without the initial zero term is a polynomial of degree 3*(k-2). This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.
%H A342980 Andrew Howroyd, <a href="/A342980/b342980.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A342980 T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(75)90050-7">Counting rooted maps by genus. III: Nonseparable maps</a>, J. Combinatorial Theory Ser. B 18 (1975), 222-259, Table VIa.
%F A342980 T(n,n+2-k) = T(n,k).
%F A342980 G.f.: A(x,y) satisfies A(x,y) = G(x*A(x,y)^2,y) where G(x,y) = 1 + x*B(x,y) and B(x,y) is the g.f. of A082680.
%e A342980 Triangle begins:
%e A342980 1;
%e A342980 0, 0;
%e A342980 0, 1, 0;
%e A342980 0, 1, 1, 0;
%e A342980 0, 1, 8, 1, 0;
%e A342980 0, 1, 20, 20, 1, 0;
%e A342980 0, 1, 38, 131, 38, 1, 0;
%e A342980 0, 1, 63, 469, 469, 63, 1, 0;
%e A342980 0, 1, 96, 1262, 3008, 1262, 96, 1, 0;
%e A342980 0, 1, 138, 2862, 12843, 12843, 2862, 138, 1, 0;
%e A342980 ...
%t A342980 G[m_, y_] := Sum[x^n*Sum[(n + k - 1)!*(2*n - k)!*y^k/(k!*(n + 1 - k)!*(2*k - 1)!*(2*n - 2*k + 1)!), {k, 1, n}], {n, 1, m}] + O[x]^m;
%t A342980 H[n_] := With[{g = 1 + x*G[n - 1, y]}, Sqrt[InverseSeries[x/g^2 + O[x]^(n + 1), x]/x]];
%t A342980 Join[{{1}, {0, 0}}, Append[CoefficientList[#, y], 0]& /@ CoefficientList[ H[11], x][[3;;]]] // Flatten (* _Jean-François Alcover_, Apr 15 2021, after _Andrew Howroyd_ *)
%o A342980 (PARI) \\ here G(n,y) gives A082680 as g.f.
%o A342980 G(n,y)={sum(n=1, n, x^n*sum(k=1, n, (n+k-1)!*(2*n-k)!*y^k/(k!*(n+1-k)!*(2*k-1)!*(2*n-2*k+1)!))) + O(x*x^n)}
%o A342980 H(n)={my(g=1+x*G(n-1, y), v=Vec(sqrt(serreverse(x/g^2)/x))); vector(#v, n, Vecrev(v[n], n))}
%o A342980 { my(T=H(8)); for(n=1, #T, print(T[n])) }
%Y A342980 Columns (and diagonals) are A006416, A006417, A006418.
%Y A342980 Row sums are A099553(n+1).
%Y A342980 Cf. A082680, A269920, A342981, A342985, A343090.
%K A342980 nonn,tabl
%O A342980 0,13
%A A342980 _Andrew Howroyd_, Apr 01 2021